Spectroscopic factor

The differential cross section is then proportional to the spectroscopic factor (or pole strength) ... 

Electron correlations show up in two ways in the measured cross sections. If the initial target state is well described by the independent particle Hartree-Fock approximation, the experimental orbital (6) is the Hartree-Fock orbital. Correlations in the ion can then lead to many transitions for ionisation from this orbital, rather than the expected single transition, the intensities of the lines being proportional to the spectroscopic factors S K... 

Figure 2. Energy level diagram depicting cases where photodissociation thresholds are determined by spectroscopic factors (left side) and thermodynamic factors (right side).

This is a very important result. It states that both dipole amplitudes from the RRPA calculation are modified by a common factor that reflects the influences of electron correlations in the initial and final ionic states which are beyond mean-field electron-electron interactions. The A0a0 2-value is called the spectroscopic factor (or the quasi-particle strength or the pole strength or the renormalization factor) and describes the weight given to the improved 2p photoionization cross section as compared to a calculation which does not include these specific electron correlations. The remaining intensity is transferred to satellite processes... 

It is clear from (11.1,11.10,11.14) that the differential cross section in the weak-coupling binary-encounter approximation is proportional to the spectroscopic factor S/(a), defined by... 

It is the centroid of the energies of the states /) of the manifold a, where the weights are the spectroscopic factors. 

The spectroscopic factors are critical quantities in determining the accuracy of a configuration-interaction calculation of the structure of the ion. The ion state /) is written in the weak-coupling representation as... 

The spectroscopic factor (11.16) is the absolute square of the coefficient of the one-hole state a) in this expansion. Determination of one coefficient in each of several eigenstates of Hi in the representation is a strong constraint on the calculation. Furthermore the determination of spectroscopic factors depends only on the validity of the weak-coupling... 

The distorted-wave impulse approximation using Hartree—Fock orbitals is confirmed in every detail by fig. 11.5, which shows momentum profiles for argon at =1500 eV. The whole experiment is normalised to the distorted-wave impulse approximation at the 3p peak. It represents the remainder of the confirmation in this case of the whole procedure of electron momentum spectroscopy. The Hartree—Fock orbitals give complete agreement with experiment for two manifolds, 3p and 3s. The spectroscopic factor Si5.76(3p) is measured as 1, since no further states of the 3p manifold are identified. Later experiments give 0.95 and this is the value used for normalisation. The approximation describes the momentum-profile shape for the first member of the 3s manifold at 29.3 eV within experimental error. The shape for the manifold sum of cross sections agrees and its... 

Fig. 11.5. The 1500 eV noncoplanar-symmetric momentum profiles for the argon ground-state transition (15.76 eV), first excited state (29.3 eV) and the total 3s manifold (McCarthy et ai, 1989). Hartree—Fock curves are indicated DWIA, distorted-wave impulse approximation PWIA, plane-wave impulse approximation. Experimental data are normalised to the 3p distorted-wave curve with a spectroscopic factor Si5.76(3p) = 0.95. The experimental angular resolution has been folded into the calculations.

Table 11.1. Spectroscopic factors for the 3s manifold of argon. EXP, McCarthy et al. (1989). The error in the last figure is given in parentheses. Target—ion, configuration interaction in the target and ion. Ion, configuration interaction in the ion only. Pert, perturbation theory...

Fig. 11.7. Noncoplanar-symmetric momentum profiles at the indicated energies for the indicated transitions in argon, compared with calculated profiles (McCarthy et ai, 1989). Experimental data are normalised to the distorted-wave impulse approximation for the summed 3s manifold. Calculations are indicated by the square of a Hartree—Fock orbital multiplied by a spectroscopic factor. Configuration-interaction curves (Cl) are described in the text.

There are states at 35.63 eV and 37.15 eV that have the 3p momentum distribution (fig. 11.7(b) and (c)). Fig. 11.7(b) includes both the 3p momentum distribution with S35,63(3p) = 0.01 and 0.67 of the cross section calculated with full correlation by Mitroy et al (1984), marked CI(/ = 1). The ground-state correlations cause a small difference in shape. The respective observed spectroscopic factors 0.01 and 0.03 for the two states agree with a number of many-body calculations of the 3p manifold. 

The valence structure of xenon is similar to that of argon in that the valence p manifolds each have one state with a spectroscopic factor near unity and the inner-valence s manifold is severely split. The additional feature of xenon is the possibility of testing relativistic calculations of the orbitals. The spin—orbit splitting of the Sp /2 and 5pi/2 manifolds can be experimentally resolved. 

Fig. 11.10. Spectroscopic factors for the noncoplanar-symmetric ionisation of xenon to the 5s5p (e=23.4 eV) ion state (lower) and the ratio of spectroscopic...

Fig. 11.13. The 1000 eV noncoplanar-symmetric momentum profiles for lead (Frost et al., 1986). Curves show the plane-wave impulse approximation. The experiment is normalised at the peak of the 6p-manifold profile (a). The 14.6 eV and 18.4 eV states of the 6s manifold are indicated by (b) and (c). Spectroscopic factors are given in table 11.2. For (a), (b) and (c) respectively the Hartree—Fock calculation (broken curve) is normalised to multiconfiguration Dirac—Fock (solid curve) by factors 0.82, 0.70 and 0.64.

The spectroscopic factors for the 6p and 6s manifolds are compared in table 11.2 with the relativistic calculations of Frost et al. (1986) that include target and ion correlations. Fig. 11.12 shows that states of the 6s manifold obey the weak-coupling approximation, so that their spectroscopic factors are momentum-independent. 

Table 11.2. Eigenvalues and spectroscopic factors for observed states of Pb (Frost et al., 1986). The spectroscopic factors for the 6p manifold are evaluated at p—0.35 a.u. (upper) and p=0.55 a.u. (lower)...

Coulomb distortion effects [14], a spectroscopic factor of 0.65 for the 3si/2 proton with occupation number 0.75 has been obtained [15,16]. 

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